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Cosparse analysis model (CAM) provides a new signal processing paradigm for recovering cosparse signals with respect to a given analysis operator from the undersampled linear measurements in the context of emerging theory of compressed sensing (CS). The sparse analysis recovery/cosparse recovery is a key one brought up by this new paradigm. In this paper, we propose a new family of analysis pursuit algorithms for the sparse analysis recovery problem when the signals obey the cosparse analysis model, termed as iterative cosupport detection estimation (ICDE). ICDE is an algorithmic framework, which alternates between detecting a cosupport set of the unknown true signal and estimating the underlying signal by solving a truncated analysis pursuit problem on the detected cosupport. Further, we propose effective implementations of ICDE equipped with an efficient thresholding strategy for cosupport detection. Empirical performance comparisons show that ICDE is favorable in comparison with the state-of-the-art sparse analysis recovery algorithms. Source code of ICDE has been made publicly available on Github: https://github.com/songhp/ICDE.